Before explaining the theory behind the LAGUNA package we present a sample calculation to show the reader what LAGUNA is able to compute. We will carry out some calculations in the group algebra of the dihedral group of order 16 over the field of two elements. First we create this modular group algebra.
gap> K := GF( 2 ); GF(2) gap> G := DihedralGroup( 16 ); <pc group of size 16 with 4 generators> gap> KG := GroupRing( K, G ); <algebra-with-one over GF(2), with 4 generators>
The group algebra KG has some properties and attributes that are direct consequences of its definition. These can be checked very quickly.
gap> IsGroupAlgebra( KG ); true gap> IsPModularGroupAlgebra( KG ); true gap> IsFModularGroupAlgebra( KG ); true gap> UnderlyingGroup( KG ); <pc group of size 16 with 4 generators> gap> LeftActingDomain( KG ); GF(2)
Since KG is naturally a group algebra, the information provided by LeftActingDomain can also be obtained using two other functions as follows.
gap> UnderlyingRing( KG ); GF(2) gap> UnderlyingField( KG ); GF(2)
Let us construct a certain element of the group algebra. For example, we take a minimal generating system of the group G and find the corresponding elements in KG.
gap> MinimalGeneratingSet( G ); [ f1, f2 ] gap> l := List( last, g -> g^Embedding( G, KG ) ); [ (Z(2)^0)*f1, (Z(2)^0)*f2 ]
Now we construct an element x as follows.
gap> a :=l[1]; b:=l[2]; # a and b are images of group generators in KG (Z(2)^0)*f1 (Z(2)^0)*f2 gap> e := One( KG ); # for convenience, we denote the identity by e (Z(2)^0)*<identity> of ... gap> x := ( e + a ) * ( e + b ); (Z(2)^0)*<identity> of ...+(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2
We may investigate some of the basic properties of our element.
gap> Support( x ); [ <identity> of ..., f1, f2, f1*f2 ] gap> CoefficientsBySupport( x ); [ Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0 ] gap> Length( x ); 4 gap> TraceOfMagmaRingElement( x ); Z(2)^0
We can also calculate the augmentation of x, which is defined as the sum of its coefficients.
gap> Augmentation( x ); 0*Z(2) gap> IsUnit( KG, x ); false
Since the augmentation of x is zero, x is not invertible, but 1+x is. This is again very easy to check.
gap> y := e + x; (Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2 gap> IsUnit( KG, y ); true
LAGUNA can calculate the inverse of 1+x very quickly.
gap> y^-1; (Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f3+(Z(2)^0)*f4+(Z(2)^0)*f1*f2+(Z(2)^ 0)*f1*f3+(Z(2)^0)*f1*f4+(Z(2)^0)*f2*f4+(Z(2)^0)*f1*f2*f4+(Z(2)^0)*f2*f3*f4+( Z(2)^0)*f1*f2*f3*f4 gap> y * y^-1; (Z(2)^0)*<identity> of ...
We may also want to check whether y is symmetric, that is, whether it is invariant under the classical involution; or whether it is unitary, that is, whether the classical involution inverts y. We find that y is neither.
gap> Involution( y ); (Z(2)^0)*f1+(Z(2)^0)*f1*f2+(Z(2)^0)*f2*f3*f4 gap> y = Involution( y ); false gap> IsSymmetric( y ); false gap> y * Involution( y ); (Z(2)^0)*<identity> of ...+(Z(2)^0)*f2+(Z(2)^0)*f2*f3*f4 gap> IsUnitary( y ); false
Now we calculate some important ideals of KG. First we obtain the augmentation ideal which is the set of elements with augmentation zero. In our case the augmentation ideal of KG coincides with the radical of KG, and this is taken into account in LAGUNA.
gap> AugmentationIdeal( KG ); <two-sided ideal in <algebra-with-one over GF(2), with 4 generators>, (dimension 15)> gap> RadicalOfAlgebra( KG ) = AugmentationIdeal( KG ); true
It is well-known that the augmentation ideal of KG is a nilpotent ideal. Using Jennings' theory on dimension subgroups, we can obtain its nilpotency index without immediate calculation of its powers. This is implemented in LAGUNA.
gap> AugmentationIdealNilpotencyIndex( KG ); 9
On the other hand, we can also calculate the powers of the augmentation ideal.
gap> s := AugmentationIdealPowerSeries( KG );; gap> s[2]; <algebra of dimension 13 over GF(2)> gap> List(s,Dimension); [ 15, 13, 11, 9, 7, 5, 3, 1, 0 ] gap> Length(s); 9
We see that the length of this list is exactly the nilpotency index of the augmentation ideal of KG.
Now let's work with the unit group of KG. First we calculate the normalized unit group, which is the set of elements with augmentation one. The generators of the unit group are obtained as explained in Chapter 3. This can be computed very quickly, but further computation with this group is very inefficient.
gap> V := NormalizedUnitGroup( KG ); <group of size 32768 with 15 generators>
In order to make our computation in the normalised unit group efficient, we calculate a power-commutator presentation for this group.
gap> W := PcNormalizedUnitGroup( KG ); <pc group of size 32768 with 15 generators>
GAP has many efficient and practical algorithms for groups given by a power-commutator presentation. In order to use these algorithms to carry out computation in the normalised unit group, we need to set up isomorphisms between the outputs of NormalizedUnitGroup and PcNormalizedUnitGroup.
The first isomorphism maps NormalizedUnitGroup(KG) onto the polycyclically presented PcNormalizedUnitGroup(PC). Let's find the images of the elements of the group G in W.
gap> t := NaturalBijectionToPcNormalizedUnitGroup( KG ); MappingByFunction( <group of size 32768 with 15 generators>, <pc group of size\ 32768 with 15 generators>, function( x ) ... end ) gap> Image(t) = W; true gap> List( AsList( G ), x -> ( x^Embedding( G, KG ) )^t ); [ <identity> of ..., f1, f2, f4, f8, f1*f2, f1*f4, f1*f8, f2*f4, f2*f8, f4*f8, f1*f2*f4, f1*f2*f8, f1*f4*f8, f2*f4*f8, f1*f2*f4*f8 ]
The second isomorphism is the inverse of the first.
gap> f := NaturalBijectionToNormalizedUnitGroup( KG );; gap> Image(f) = V; true
For example, we may calculate the conjugacy classes of the group W, and then map their representatives back into the group algebra.
gap> cc := ConjugacyClasses( W );; gap> Length( cc ); 848 gap> Representative( cc[ Length( cc ) ] ); f1*f2*f3*f6*f10*f13 gap> last^f; (Z(2)^0)*<identity> of ...+(Z(2)^0)*f1*f2+(Z(2)^0)*f1*f3+(Z(2)^0)*f1*f4+(Z(2)^ 0)*f2*f3+(Z(2)^0)*f1*f2*f3+(Z(2)^0)*f1*f3*f4
Having a power-commutator presentation of the normalised unit group, we may use the full power of the GAP functionality for such groups. For example, the lower central series can be calculated very quickly.
gap> LowerCentralSeries( W ); [ <pc group of size 32768 with 15 generators>, Group([ f4*f8, f5*f7*f11*f13*f15, f6*f7*f9*f11*f13*f14*f15, f8, f9*f13, f10*f11, f12*f13, f13*f15, f14*f15 ]), Group([ f8, f9*f15, f10*f11, f12*f15, f13*f15, f14*f15 ]), Group([ f12*f15, f13*f15, f14*f15 ]), Group([ <identity> of ... ]) ]
Let's now compute, for instance, a minimal system of generators of the centre of the normalised unit group. First we carry out the computation in the group which is determined by the power-commutator presentation, then we map the result into our group algebra.
gap> C := Centre( W );; gap> m := MinimalGeneratingSet( C ); [ f8*f13*f14*f15, f13*f14*f15, f8*f12*f14*f15, f15, f4*f6*f8*f13 ] gap> List( m, g -> g^f ); [ (Z(2)^0)*<identity> of ...+(Z(2)^0)*f3+(Z(2)^0)*f1*f2+(Z(2)^0)*f3*f4+(Z(2)^ 0)*f1*f2*f3+(Z(2)^0)*f1*f2*f4+(Z(2)^0)*f1*f2*f3*f4, (Z(2)^0)*f3+(Z(2)^0)*f4+(Z(2)^0)*f1*f2+(Z(2)^0)*f3*f4+(Z(2)^0)*f1*f2*f3+( Z(2)^0)*f1*f2*f4+(Z(2)^0)*f1*f2*f3*f4, (Z(2)^0)*<identity> of ...+(Z(2)^ 0)*f1+(Z(2)^0)*f3+(Z(2)^0)*f1*f2+(Z(2)^0)*f1*f3+(Z(2)^0)*f1*f4+(Z(2)^ 0)*f3*f4+(Z(2)^0)*f1*f2*f3+(Z(2)^0)*f1*f2*f4+(Z(2)^0)*f1*f3*f4+(Z(2)^ 0)*f1*f2*f3*f4, (Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f3+(Z(2)^0)*f4+(Z(2)^ 0)*f1*f2+(Z(2)^0)*f1*f3+(Z(2)^0)*f1*f4+(Z(2)^0)*f2*f3+(Z(2)^0)*f2*f4+( Z(2)^0)*f3*f4+(Z(2)^0)*f1*f2*f3+(Z(2)^0)*f1*f2*f4+(Z(2)^0)*f1*f3*f4+(Z(2)^ 0)*f2*f3*f4+(Z(2)^0)*f1*f2*f3*f4, (Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f3+( Z(2)^0)*f4+(Z(2)^0)*f1*f3+(Z(2)^0)*f1*f4+(Z(2)^0)*f3*f4+(Z(2)^ 0)*f1*f3*f4+(Z(2)^0)*f2*f3*f4 ]
We finish our example by calculating some properties of the Lie algebra associated with KG. This example needs no further explanation.
gap> L := LieAlgebra( KG ); #I LAGUNA package: Constructing Lie algebra ... <Lie algebra of dimension 16 over GF(2)> gap> D := LieDerivedSubalgebra( L ); #I LAGUNA package: Computing the Lie derived subalgebra ... <Lie algebra of dimension 9 over GF(2)> gap> LC := LieCentre( L ); <Lie algebra of dimension 7 over GF(2)> gap> LieLowerNilpotencyIndex( KG ); 5 gap> LieUpperNilpotencyIndex( KG ); 5 gap> IsLieAbelian( L ); false gap> IsLieSolvable( L ); #I LAGUNA package: Checking Lie solvability ... true gap> IsLieMetabelian( L ); false gap> IsLieCentreByMetabelian( L ); true
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